Test 3 math 4100, open book/pdf by Axler and/or 20 sheets of notes.
ONLY the LMS webpage can be open on your electronic devices. Proofs must
conform to level of rigor of the textbook. All solutions must be submitted
in person in classroom on LMS and on time except by written permission.
No questions to proctor will be answered during exam. Your submission to
LMS is governed by the honor code as per the RPI student handbook.
1. Answer the following questions (2 pts each) on finite dimensional vector
spaces with YES (True) or No (False): (no work needed and no partial credit)
For the next 4 questions, Let linear T:C3→C3be normal:
(i) Thas three eigenvalues.
(ii) Thas at least one 1D eigenspace.
(iii) Tcan be put in upper-triangular form.
(iv) The adjoint T∗has a two-dimensional range.
2. Answer the following questions (2 pts each) on finite dimensional vector
spaces with short answers: (no work needed and no partial credit)
For the next 4 questions, Let linear mapping T:R3→R3be self adjoint:
(i) Let v be an eigenvector of T. Calculate the inner product of T v and
v.
(ii) Give the most complete description of the matrix of T (such as its 9
entries) in the standard basis.
(iii) If the row sums and column sums of matrix of T are all ones, Calculate
explicitly a real eigenvector of Tand its corresponding eigenvalue.
(iv) Supposing that T is the identity, Calculate in terms of T, the action
of the dual T0on a linear functional φ(x, y, z)t=<(1,1,1)t,(x, y , z)t>; give
the result on the vector (1,2,3)t.
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